TLDR: Quantum mechanics (a mathematical theory of how physical particles behave) predicts that computers using quantum entanglement can solve some problems more efficiently than classical (conventional) computers can. This is a current hot topic in math and physics research.
Some configurations of quantum circuits wlil have this efficiency gain, while other configurations will be like ordinary computers. The property that gives the extra efficiency has been cutely called "magic" in some of the quantum computing literature. It has nothing to do with Harry Potter. It's just a metaphorical term for a mathematical property of some arrangements of entangled particles. Of course it's a cool property, but the name "magic" being used in the titles of some papers, perhaps makes those papers sound more interesting than they really are, to those of us not very familiar with the subject.
Maybe I'm missing something, but I'm not sure what we're really supposed to learn by seeing these papers linked here. A good intro to quantum computing for laypeople is "Quantum Computers Since Democritus" by Scott Aaronson. I've read some parts of it and it looks to me like a preferable place to start.